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A soil data set can be dense and precise yet still overrepresent places that looked interesting, were easy to reach or were sampled after an earlier high reading. Preferential-sampling models make the site-selection mechanism part of the inference instead of treating locations as fixed and harmless.
Open the chapter contents · Soil Science root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · See the selection mechanism
Route 2 · Work the bias
Route 3 · Fit and stress the correction
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · SEE THE SELECTION MECHANISM
1. When the map influences where the next sample appears
Preferential sampling occurs when the process choosing observation locations is stochastically related to the spatial process being measured. A field team may revisit high-contamination sites, sample promising ore zones more densely, or avoid steep terrain. The resulting locations contain information about selection, not merely geography.
Conventional geostatistics often conditions on locations as if they were fixed independently of the latent soil surface. That can be misleading when the same unobserved surface influenced site choice. The solution is not to discard every legacy observation. It is to state a location model and examine the outcome and site processes jointly.
This extends the Soil Science root and the spatially balanced sampling guide. Design prevents future bias; joint modelling investigates bias already embedded in the site history.
CHAPTER 2 OF 12 · SEE THE SELECTION MECHANISM
2. A joint location–outcome model
Let the soil outcome at location s be Y(s) = μ + S(s) + ε(s), where S(s) is a spatial latent process and ε(s) is measurement or microscale error. A simple preferential location model treats sampling sites as a point process with log intensity log λ(s) = α + γS(s) + q(s)ᵀδ.
The coefficient γ connects the unobserved soil surface to sampling intensity. If γ > 0, higher latent values attract more sites after accounting for observed site-selection covariates q(s). If γ = 0, the model says site intensity is conditionally independent of S(s).
Inference integrates over the shared latent field in the joint likelihood p(y, locations) = ∫p(y|S, locations)p(locations|S)p(S)dS. That integral is computationally demanding, and the result depends on both submodels.
CHAPTER 3 OF 12 · WORK THE BIAS
3. Worked eight-cell landscape
Consider eight equal-area cells with invented latent soil values from 2 to 9. Their population mean is 5.5. Suppose the relative chance of a legacy site is proportional to exp(0.2 × value). The higher cells are more likely to be represented.
| Cell | Latent value | Relative site intensity exp(0.2 × value) |
|---|---|---|
| 1 | 2 | 1.492 |
| 2 | 3 | 1.822 |
| 3 | 4 | 2.226 |
| 4 | 5 | 2.718 |
| 5 | 6 | 3.320 |
| 6 | 7 | 4.055 |
| 7 | 8 | 4.953 |
| 8 | 9 | 6.050 |
The intensity-weighted mean is about 6.507, one unit above the equal-area mean. This does not prove a particular legacy survey is biased. It shows the direction a positive selection–surface link can create before measurement error and spatial interpolation enter.
CHAPTER 4 OF 12 · WORK THE BIAS
4. Observed covariates may explain the apparent preference
Suppose high values occur mainly in valley soils and roads also follow valleys. A dense road-side sample can look preferential with respect to Y, yet the dependence may disappear after conditioning on road distance, landform and access.
This distinction matters. If observed q(s) explains site choice, a design-weight or covariate-adjusted analysis may be sufficient. If the latent residual S(s) still influences selection, conditioning only on q(s) leaves informative location bias.
Build a site-selection ledger from field notes, maps, commissioning rules and dates. “Unknown” is a legitimate category. It is safer than inventing a random-sampling history because the coordinates happen to cover the map.
CHAPTER 5 OF 12 · WORK THE BIAS
5. The joint correction changes both surface and uncertainty
A joint model uses the absence and concentration of sites as evidence about S(s), through the specified location process. With γ > 0, a cluster of sites can partly signal a high latent surface rather than eight independent confirmations. Unsampled regions may receive different predictions and wider uncertainty than under a location-ignoring fit.
| Model comparison | Location process | Question answered |
|---|---|---|
| Ordinary geostatistics | Conditions on observed sites | What surface fits values at these fixed locations? |
| Covariate-adjusted selection | Uses observed access/design variables | Is selection ignorable after known causes? |
| Joint preferential model | Shares latent S(s) with site intensity | How do values and site pattern inform one another? |
Compare posterior or likelihood profiles for γ, maps, population averages and exceedance areas. A small map difference does not mean the mechanism is absent; it may mean the available data cannot identify it strongly.
CHAPTER 6 OF 12 · FIT AND STRESS THE CORRECTION
6. Combine legacy sites with a probability sample
A new spatially balanced probability sample is valuable because its selection mechanism is known. Fit the legacy and new data with source indicators, measurement harmonisation and explicit sampling roles. The new panel can help estimate broad status; legacy sites can preserve trend or local detail where their comparability is defensible.
Do not silently pool depths, laboratory methods or support sizes. A 0–10 cm composite and a point reading at 15 cm are not exchangeable simply because their coordinates are close. Measurement harmonisation comes before sophisticated location correction.
Use held-out probability sites to assess predictions from legacy-only and combined models. This is a stronger test than randomly splitting a clustered legacy set, which can leak nearby spatial information into validation.
CHAPTER 7 OF 12 · FIT AND STRESS THE CORRECTION
7. Changed context: access removes steep terrain
Imagine a later campaign that excludes slopes above 25 degrees for safety. Site scarcity on steep terrain now has an observed explanation. If the target population still includes those slopes, the design has a coverage gap; if they are formally outside the target, the estimand changes.
The joint model should not interpret every empty steep cell as evidence of low soil values. Include the access mask or define zero sampling intensity outside the reachable frame. Then report where predictions are extrapolations beyond sampled support.
Connect this to the area-of-applicability guide. Preferential correction addresses informative site placement; applicability analysis asks whether prediction covariates resemble the training domain. Both can matter in the same cell.
CHAPTER 8 OF 12 · FIT AND STRESS THE CORRECTION
8. Identification requires sensitivity analysis
The latent field drives both observations and locations, so γ can trade off against spatial range, variance, trend and omitted site covariates. A flexible intensity surface may absorb the very dependence γ was meant to identify; an overly rigid one may exaggerate it.
Fit plausible alternatives: different trend bases, access covariates, spatial ranges, point-process resolutions and priors. Compare not only γ but the decisions—mean stock, exceedance area, priority zones and uncertainty.
If site dates are available, model sequential selection where earlier observations influenced later visits. A single static point process can blur this feedback. The correct history may be “inspect, observe, revisit”, not one simultaneous draw.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Clustering is not automatically preferential. Sites can cluster because of logistics, stratification or a legitimate target subpopulation. Preferential sampling refers to dependence with the process of interest after the stated conditioning.
A joint model does not reconstruct an unknowable sampling history with certainty. Its correction is model-based and should be presented with sensitivity ranges. Design weights from a genuine probability survey and latent-process corrections answer different problems; do not mix their interpretations casually.
Coordinates may also be displaced for privacy or rounding, and soil observations can have depth, support and laboratory differences. A location model cannot repair incompatible measurements. Professional environmental or remediation decisions require domain-specific standards and field verification.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Evidence and reproducibility route
Diggle, Menezes and Su define preferential sampling, derive a joint likelihood and show that ignoring the mechanism can materially alter inference: Geostatistical inference under preferential sampling. Stevens and Olsen develop spatially balanced probability sampling in Spatially balanced sampling of natural resources.
Archive the target population, site dates, original selection rules, failed and replacement sites, access masks, laboratory methods, coordinate transformations, covariates, outcome and intensity equations, priors, mesh or grid settings, and code. Preserve a location-only diagnostic as well as the outcome fit.
Map where conclusions depend on the preferential mechanism. A single corrected mean hides the regions in which site selection changed the evidence most.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does a random train–test split detect preferential sampling?
Usually not reliably. Both sets can inherit the same site-selection pattern and spatial leakage.
Should legacy data be deleted?
No. They may hold valuable history and local detail. Their role and selection mechanism should be explicit.
Is γ a direct measure of bias in the final map?
No. It is a model parameter linking latent surface and intensity. Map impact also depends on other parameters, data and prediction targets.
Can observed access covariates solve the problem?
They can make selection conditionally ignorable if they adequately explain it. That is an empirical and substantive assumption to test.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Final transfer check and connected reading
Recalculate the eight-cell equal-area and intensity-weighted means. Then change the intensity rule to exp(−0.2 × value): the weighted mean should move below 5.5. Finally add a road covariate that explains the site pattern and ask whether γ is still needed. Each change tests a different causal story.
Continue through the Soil Science root, spatially balanced sampling, spatial cross-validation and area-of-applicability masks. Together they separate how sites entered the evidence, how predictions are tested and where transfer is credible.
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